The Reciprocation Problem

The “reciprocation problem”: a mathematical tragedy in relationships regarding how often people should ask each other to hang out

The Setup

Person X and person Y are friends (or lovers or close work colleagues or whatever). Person X and Person Y happen to both feel the same way about each other (i.e., equal amounts of interest, affection, lust, respect, etc.)
Person X’s ideal is to make plans with person Y every two weeks, whereas person Y (who has a lower amount of free time, or less need for social interaction, or a project they are prioritizing, or whatever) wants to see person X every three weeks. Hence, they differ in their preferred time interval between hangouts.

So what happens? Approximately every two weeks, person X asks person Y to spend time together, which means that person X ends up doing essentially 100% of the invites (since three weeks rarely elapse without Y receiving an invitation from X, so Y almost never asks X to spend time together).†

In other words, Y wants to see X at only a moderately different rate than X wants to see Y (e.g., every three weeks instead of every two weeks) but ends up doing 0% of the invitations.

Person X then assumes that their relationship is imbalanced, and person Y must not feel the same way about the relationship that they do (but happens to be wrong). This can lead to awkwardness and relationship problems.

So how should X handle a situation where X ends up doing all of the inviting without Y reciprocating (but with Y agreeing to see X whenever X does send an invite)? We’ll assume in each of these cases that the people actually do spend time together when an invite is made (i.e., it is not a case of one person purposely ignoring another).

Some would suggest that, if both people simply ask each other their preferences, that can work best for the right types of people. Especially if both are on board with such explicitness of conversations about relationships, know the other person is on board too, and are confident that negative ramifications such as damaging awkwardness won’t result from that explicitness. In most cultures, the explicitness of the form “I’d prefer to see you every three weeks, how often do you want to see me?” is not the norm but certainly is normal in some subcultures.

Always Ask Strategy

The default strategy would be for person X to just persist in making the invitations every two weeks. One drawback is that person X might feel bad about always being the one to make invitations. Another drawback is that person Y might feel frustrated because they end up having to stall regularly on those invites to get their desired rate of spending time together, or else agree to see X on X’s preferred schedule rather than their own. Yet another drawback is that person X might be misreading the signs: maybe person Y just feels bad about saying no and so agrees to see X despite not wanting to? Person X knows that person Y is not reciprocating the invitations by sending invites back to X but doesn’t know the reason Y isn’t reciprocating.

Tit-For-Tat Strategy

Another strategy would be for person X to never make two invitations in a row. That means that person X would make the first invite (after two weeks), and then person Y would make the next invite (after three weeks) and then person X the next and so on. This isn’t a terrible solution since they would see each other every 2.5 weeks, which is a nice compromise. However, it does have a very major drawback, which is that if person Y forgets to make an invite back, then they’ll be stuck not seeing each other. In other words, it’s leaves room for mistakes, and could inadvertently destroy a great relationship. Hence far from ideal! This could be expanded to a “Tit-For-Two-Tats” strategy, where X makes two invites in a row but not more than two. But this actually is not really more robust in this scenario than Tit-For-Tat, since after two invites from X (which will definitely happen in this scenario), if Y then forgets to make the next one, no more invitations will occur.

Exponential Strategy

A third strategy would be for person X to double their invite time interval each time their last invite does not get an invite in return, and then reset back to their original invite time as soon as they get an invite back. (It doesn’t have to be double, of course, it could be X multiplying the invite time by any constant C>1.)

To see what I mean in more detail, consider the situation where X still prefers to see Y every two weeks, but now Y prefers to see X every ten weeks. First, person X makes an invite after two weeks, and they see each other at that time. Then since Y doesn’t reciprocate, X multiplies their time by two and so makes an invite after four weeks. Since Y again doesn’t reciprocate, X multiplies their time by two again and makes an invite after eight weeks. Since person Y’s desired time to see each other is ten weeks, then Y will end up making the next invite (since X wouldn’t make their next invite for 16 weeks). Now since a reciprocation occurs, X resets and so sends the next invite in 2 weeks, then the invite after that in 4 weeks, then eight weeks, etc.

There are some neat things about this strategy. First of all, it’s robust to mistakes since even if person Y accidentally forgets to make a reciprocation, person X will still end up reaching back out with an invite. Second, it does a pretty good job of balancing the desires of both parties by finding an average meeting frequency that is a compromise of both their ideals. The math gets a bit complicated†† but, suffice it to say, for most values of K (person X’s ideal timing between invites), C (the multiplier that X applies to their invite time interval with each unreciprocated invite), and N (person Y’s ideal timing between invites), the average time delay of the two people seeing each other will be fairly close to halfway between X’s and Y’s ideal time delay (to be more precise, it’s usually in the range of 25% of the way from K to N up to about 75% of the way from K to N).

Another neat thing about this strategy is that in the event that X has misjudged this situation, and Y actually doesn’t want to spend time together, Y gets pestered with exponentially decreasing frequency, meaning that the total annoyance Y experiences and the total embarrassment from non-reciprocation that X experiences are both limited.

One final point about the exponential strategy is that it works well if both parties use it even in an environment where forgetting is common (i.e., if 30% of the time people get distracted and so forget to make an invite).

Takeaways

In reality, one would, of course, not do calculations like this formally, and this simple model doesn’t include all relevant factors. But perhaps the Exponential Strategy can give us a decent intuition for how we might handle these sorts of situations in real life. When we want to see someone at regular intervals, and the person does agree to see us when we make invites but doesn’t make invites of their own, increase our invite time interval by some constant factor (say, 2) each time we have a non-reciprocation, but reset back to our most desired invite time interval whenever a reciprocation occurs.

† Note that this math works out even if X has some amount of noise in when they ask Y to spend time together, but as this noise gets large, the math eventually breaks down.

†† The average delay D that occurs between invites when person X is using the Exponential Strategy will be:
D = (N + K * sum_{t=0}^{M-1} C^t ) / (1 + M) where M =Ceiling[Log[N/K]/Log[C]]